Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let k be a field and let R=k[X1,…,Xn] be a polynomial ring. Let F⊂R be a finite subset generating a radical ideal I with precisely e solutions over an algebraic closure of k. Is there a monomial order on R with a Groebner basis of degree ≤e for I? Answer AttributionSource : Link , Question … Read more

Gröbner bases of resultants and their monomial ideals

$\newcommand{QQ}{\mathbb{Q}}$ Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial $$ f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 – i} $$ Now let $$ g_j = \mathrm{resultant}_x(f, \frac{d^{m+1-j}}{dx^{m+1-j}} f) $$ for $j=1,\ldots,m$ and let $$ I_s = (g_1,\ldots, g_s) $$ be an ideal of $R$ (and of $S=\QQ[a_1,\ldots,a_m]$). … Read more

Compute equalizer of maps of polynomial rings, perhaps using Gröbner bases

Suppose that k is a field and I have two ring homomorphisms ϕ,ψ:k[x1,…,xm]→k[y1,…,yn]. How can I use Gröbner bases (or other computational tools) to compute the subring of elements a such that f(a)=g(a)? Comments: When I say “compute”, I mean actually compute, for example using Sage, Macaulay2, etc. I actually don’t care about using Gröbner … Read more

Algorithm to detect if an element of a (commutative) ring is in a subring?

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring. Is there an analogous story for subrings — i.e. given a ring (finitely-generated over a field k, say), and a finite set … Read more

How can I include irreducibility in a Groebner basis calculation?

I’m trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation qn=mf, where each of the variables refers to a polynomial in some ring, say Q[x]. If I know that f is irreducible, and that f is coprime to both q and n, then the equation … Read more

Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials p1(x1,x2,…,xn)=0,p2(x1,x2,…,xn)=0,⋮pm(x1,x2,…,xn)=0,(x1−l1)(x1−(l1+1))(x1−(l1+2))⋯(x1−(u1−1))(x1−u1)=0,⋮(xn−ln)(xn−(ln+1))(xn−(ln+2))⋯(xn−(un−1))(xn−un)=0⋯(xn−un)=0. Here l1,u1,…,ln,un are non-negative integers. The last n equations mean that xi∈{li,li+1,li+2,…,ui−1,ui} for all i∈{1,…,n}. I know that we can use Groebner bases to solve the polynomial system. But is there a better way to solve such over-determined polynomial system? I will … Read more

Can a minimal generating set for an ideal always be made into a Groebner basis?

Let I⊆k[x0,…,xn] be an ideal, generated by some polynomials F1,…,Fr, all homogeneous and of the same degree. Suppose r is the smallest number of generators that will suffice to generate the ideal. Can one choose a monomial order or change coordinates to ensure that this generating set is also a Groebner basis for the ideal? … Read more

Systems of polynomial equations

Hi all, I’m an engineer assigned to determine some parameters of a manipulator (ie., calibration). It has a number of parameters, but after some manipulations of its dynamic equations, I can have the following equations: lTiRxRip=0 with i=0,..,N (1) where li (3×1 vector) and Ri (3×3 vector) can be measured and computed using external devices. … Read more

Numerical solution for a system of multivariate polynomial equations

Hi all, I have a system of 6th-order polynomial equations in 4 variables q1,q2,q3,q4 (i.e. polynomials with all the terms such as q61,q62,q42q23): Pk(q1,q2,q3,q4)=0 with k=2,…,N I don’t have any good guess of the q_i. So, Newton method and its variant won’t work because they need a good starting point to converge to the right … Read more

Lower bounds on the degrees of representatives of unu^n as n→∞n \to \infty

Let k be an algebraically closed field and A a finitely generated k-algebra, together with a specified surjective morphism ϕ:k[x1,…,xn]→A. For f∈A, define deg(f) to be the minimum of deg(g), where g ranges over all polynomials in k[x1,…,xn] such that ϕ(g)=f. [Note: by deg(g), I mean the degree of the highest-degree monomial, where deg(xi11⋯xinn)=i1+⋯+in.] If … Read more